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Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $...

Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently? $$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$

I'm interested in finding the expected rank of some random matrix $A$ (I don't want to specify its distribution right now, since my question makes sense in general). Computing $\mathbb{E} \ \mathrm{...